Vectors

Vectors PreCalculus Presentation by:

History - John Klein Review - Brad Parbs Solving a problem - Sam Weber Applications - Mike Zabrowski Works Cited

avaiable online at BradParbs.com/vectors

pg. 2 pg. 3 pg. 4 pg. 5 pg. 6

2 Vectors were first thought about by Isaac Newton in his book, Principia Mathematica written in 1687. In the Principia Mathematica, Newton dealt extensively with what are now considered vectorial entities, like velocity and force, but never the concept of a vector. The systematic study and use of vectors were a 19th and early 20th century phenomenon. Vectors were born in the first two decades of the 19th century with the geometric representations of complex numbers. Caspar Wessel, Jean Robert Argand, and Carl Friedrich Gauss all first conceived the idea of complex numbers as points in the two-dimensional plane. Mathematicians and scientists worked with and applied these new numbers in various ways; for example, Gauss made crucial use of complex numbers to prove the Fundamental Theorem of Algebra. In 1837 and William Rowan Hamilton showed that the complex numbers could be considered abstractly as ordered pairs (a,b) of real numbers. This idea was a part of the campaign of many mathematicians, including Hamilton himself, to search for a way to extend the two-dimensional “numbers” to three dimensions; but no one was able to accomplish this, while preserving the basic algebraic properties of real and complex numbers. After a good deal of frustration, in 1843 Hamilton was finally inspired to give up the search for such a three-dimensional number system and instead he invented a four-dimensional system that he called quaternions. Hamilton had been knighted in 1835, and he was a wellknown scientist who had done fundamental work in optics and theoretical physics by the time he invented quaternions, so they were given immediate recognition. In turn, he devoted the remaining 22 years of his life to their development and promotion. He wrote two exhaustive books, Lectures on Quaternions (1853) and Elements of Quaternions (1866), detailing not just the algebra of quaternions but also how they could be used in geometry. At one point, Hamilton wrote, “I still must assert that this discovery appears to me to be as important for the middle of the nineteenth century as the discovery of fluxions was for the close of the seventeenth.” He acquired a disciple, Peter Guthrie Tait (1831--1901), who in the 1850s began applying quaternions to problems in electricity and magnetism and to other problems in physics. In the second half of the 19th century, Tait’s advocacy of quaternions produced strong reactions, both positive and negative, in the scientific community. (Vectors) At about the same time that Hamilton discovered quaternions, Hermann Grassmann wrote The Calculus of Extension. In this book he studied the theory of tides. Grassmann expanded the conception of vectors to an arbitrary number. Then, Grassmann received very little credit for his discovery due to his obscure style and because he was a school teacher without a major scientific reputation. The development of the algebra of vectors and of vector analysis as we know it today was first revealed in sets of remarkable notes made by J. Willard Gibbs for his students at Yale University. He concluded that vectors would provide a more efficient tool for his work in physics. So, beginning in 1881, Gibbs privately printed notes on vector analysis for his students, which were widely distributed to scholars in the United States, Britain, and Europe. The first book on modern vector analysis in English was Vector Analysis in 1901.Gibbs’s notes were assembled by one of his last graduate students, Edwin B. Wilson. Vectors are now the modern language of a great deal of physics and applied mathematics and they continue to hold their own intrinsic mathematical interest.

3 Throughout the study of Pre Calculus, vectors are studied for quite some time. Also, in Physics, an understanding of vectors is essential. Basically, there are a few different attributes and features of vectors that need to be understood, then you can move onto more complex problems involving vectors. Many problems in Physics as well as Pre Calc are solved using vectors. We write a vector by writing the initial point as a letter, then the end point also as a letter, then draw a half arrow on top. To show magnitude, we write the magnitude, then surround it with two lines on both sides. The magnitude is also the length. Vectors are very similar to line segments when written, and we need to take care not to confuse them. There are two things to remember about a vector, it is made up of a size and a direction. The only difference between different vectors is going to be the size or the direction. When a vector is at an angle, one can easily break it down into two components, both parallel to either the x or y axis. When we break it down, we need to make sure to look at the angles and sizes, to make sure no mistakes are made. If we break the vector down, and we end up with a vector having a magnitude of one, that vector is a unit vector, used only for direction. This component form, when broken down, can also be written out in brackets. We put the vector at (0,0) and we write the end point in brackets. <3,3> or <5,2>. This allows us to easily do mathematics and solve for different things. To aid in solving problems, we use unit vectors. Unit vectors have a magnitude, that is, the size, of one. Magnitude is basically the power of it. Unit vectors help to indicate direction. A key concept when using vectors, is the mathematics involved. In the past, we have learned addition and scalar multiplication. Addition is quite simple, we simply place the two vectors together, and draw in a third vector from the starting point of the first vector to the end point of the second vector. The size and direction of this resulting vector is the answer, or simply the resultant. Scalar multiplication is also quite simple, once you know how to. Scaling a vector is multiplying it by a certain factor. We take the vector in the bracketed form, then multiply each number by the scale. A vector going to <2,2> scaled by 2 would end up at <4,4>. We can also break a vector into a linear combination form. This allows us to use the horizontal and vertical components easier. We take the first number, put that in front of i, the second number in front of j. We can easily add vectors in linear form, as well as finding the angle between them. To find the angle, we take the tangent of the first number of the first vector over the first number of the second vector. We also take the tangent of the second numbers, then add those both together. The dot product is also used often, and is found by taking the magnitude of two vectors, multiplying them together, also multiplying the cos of the angle between them.

4 F1= 65N at 70° F2= 55N at 20° 1) Draw the components out and figure out sin x and sin y. then find out cos x and cos y. By drawing the components we find out which values we need to find when using sin and cos. For sin we need to opposite and hypotenuse side. For cos we need the adjacent side and hypotenuse side. By finding the values we then know what we are solving for. 2) Once you draw the components and find out both sin and cos of x and y you need to find their sums. You find the values of the numbers and then add. You need to multiply the number in Newtons by cos of the degree of the 55 cos (20) vector. 55 sin (20) + 65 cos (70) + 65 sin (70) + 74 + 80

3) After adding your sin and cos values you have found the components of the resultant. The components of the resultant are used to find the magnitude. The resultant <80,74> 4) To find the magnitude, which is the size of the vector, you need to square each component of the resultant and then find the square root of that sum. √ 802 + 742

= 108.98

5) Finding the magnitude resembles using the Pythagorean Theorem. You can also find components if the magnitude is given along with another component. For example if the given magnitude is 108.98 and 80 as the component. You start by squaring the magnitude. 108.982

Then you get 11876.6404.



Then you square the component 80.

Then you get 6400 Then you subtract the magnitude squared and the component squared. 11876.6404-6400 = 5476.6404 Then you need to find the square root of the difference. √ 5476.6404 After those steps you come back to 74, which was the other component.

5 As we know, a vector is anything with a size and a direction. As such, there are many examples of vectors in everyday life. First, loot at a moving car. A car moves at a certain speed (in this case, the size) and moves in a direction. Therefore, a car travelling down a road is a vector. There are many other examples of vectors in the world of sports. For example, when a soccer ball is kicked it usually moves in a straight path—size (speed) and direction. With the boot of a skilled soccer player, however, this kick can be made into a curve. All of the different forces acting on the ball--such as the force of the kick that is brought on by the person and the spin of the ball in the air--deflects the ball’s path and results in a curved flight pattern. The concept of a ball moving at a specific velocity through the air or ground is common in many other sports as well. Other examples include: a football thrown by Brett Favre, a curveball thrown by an MLB pitcher, a slap shot from a hockey player on a puck, and even a bowling ball thrown down an alley. Also, vectors can be seen within other things apart from sports. They can also be seen in the world of computers. When a person wants to tell the computer to make an image, he could do so by using pixels. Pixels could be used by telling the computer exactly what color every single individual pixel is. However, this is not the best way to do this. Making a visual by this method takes up a lot of memory, and when the angle is moved the picture gets distorted. It is much easier to use vectors when creating this image. To use vectors, one can tell the computer what direction sunlight or wind is coming from. Both sunlight and wind have a direction and a size of how strong the wind gust is or how intense the sun is beating down. When used like this, the color of the picture can be calculated by the computer to give it realistic lighting and wind effects. There are also many other uses that a vector has. For example, meteorologists use vectors to predict weather patterns. They use the wind speed and the direction that it is travelling to give a rough estimate of how the weather will be in a certain area. Vectors also can be used when sailing in a sailboat. The wind pushes the boat with a force and the boat moves in a certain direction. Vectors are so simple that they can be found in many aspects of life. Once again, anything with size and direction is a vector, so any moving object can be a vector.

6 Works Cited “Basic Vector Operations.” Basic Vector Operations. 18 May 2009 . “Vector -- from Wolfram MathWorld.” Wolfram MathWorld: The Web’s Most Extensive Mathematics Resource. 21 May 2009 . “Vectors: Fun activities with vectors.” Division of Engineering Welcome. 27 May 2009 . “Vectors.” Mathematics and Statistics | The Department of Mathematics and Statistics. 27 May 2009 . “Vectors.” The Math Forum @ Drexel University. 22 May 2009 . “Vectors.” The Physics Department - Mechanics, Vectors. 17 May 2009 .